Question

If cotθ=3/4 and the terminal point determined by θ is in quadrant 3, then: (choose all that apply)

cosθ=-3/5
tanθ=4/3
sinθ=3/5
cscθ=-5/3

Answer 1

Tan theta = 4/3 and cosine theta = -3/5 are the correct options.

Explanation:

Answer 2

With
`\theta` in quadrant 3, we should expect both
`\cos\theta` and
`\sin\theta` to be negative, so that
`\tan\theta` is positive. The corresponding reciprocal expressions
`(\sec\theta,\csc\theta,\cot\theta)` will have the same sign.

`\cot\theta=\frac34\implies\tan\theta=\frac43`

Recall that
`1+\tan^2\theta=\sec^2\theta`, which means

`\sec\theta=-√(1+\tan^2\theta)=-\frac53`

`\implies\cos\theta=-\frac35`

Also recall that
`\cos^2\theta+\sin^2\theta=1`, so

`\sin\theta=-√(1-\cos^2\theta)=-\frac45`

`\implies\csc\theta=-\frac54`

Only the first two options are correct.